Consider a one-dimensional time-independent Schrodinger equation for an electron in a double quantum well separated by an additional barrier. The potential is given by

in which * *is a Dirac delta function, , and *a* are some parameters, and *x *is the coordinate. You may find it useful to work in units *h = *1 and *m* = 1/2 and to introduce *k *defined as *E =*—k^{2}, where *E < *0 is the energy of a bound state, so that *k *is real.

**2.1** Find algebraic equations determining the energies (or *k*-values) of electron bound states for and arbitrary real 0 (positive or negative). How many bound states do you expect in this system? Describe the symmetry of their wave functions in terms of even and odd solutions.

**2.2** Calculate the critical value *a1* of the parameter *a*, at which an odd solution disappears.

**2.3** Calculate the critical value *a2* of the parameter *a*, at which an even solution disappears.

**2.4** For β > 0, consider the limits of very large and very small *a* working out the analytic expressions for *E *(or *k) *for all bound states in this system, as well as their energy (or *k) *splittings at large a. Give a sketch of your result showing *E *(or *k) *versus a for all bound states in the system.

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